Final answer:
The probability that a randomly selected monthly payment is more than $1000 is approximately 0.5014. The probability of a monthly payment between $800 and $1100 is approximately 0.0260. The probability that the sample mean of 30 mortgage payments is between $950 and $1050 is approximately 0.
Step-by-step explanation:
To find the probability that a randomly selected monthly payment is more than $1000, we need to standardize the value using the formula z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Plugging in the values, we have z = (1000 - 982) / 5180 = 0.0035. Using a standard normal distribution table, we can find that the probability is approximately 0.5014.
For the probability of a monthly payment between $800 and $1100, we need to standardize the values of $800 and $1100. Using the formula, we find that z1 = (800 - 982) / 5180 = -0.035 and z2 = (1100 - 982) / 5180 = 0.035. Again, using the standard normal distribution table, we can find that the probability is approximately 0.5138 minus 0.4878 = 0.0260.
To find the probability that the sample mean of 30 mortgage payments is between $950 and $1050, we need to standardize the values of $950 and $1050 and use the formula z = (x - μ) / (σ / sqrt(n)), where n is the sample size. Plugging in the values, we have z1 = (950 - 982) / (5180 / sqrt(30)) = -0.28 and z2 = (1050 - 982) / (5180 / sqrt(30)) = 0.28. Using the standard normal distribution table, we can find that the probability is approximately 0.39 minus 0.39 = 0.